The terms outside the bracket is the same order as the
differential of the term inside the bracket.

Let u=

Observe how the ‘cancels’

When the power of x is at most 1 both outside and inside the
bracket

Let u=3x+2

Rearrange to find x in terms of u.

Substitute this in for x

Note at the end that

The numerator is the same order as the differential of the
denominator.

Let

Observe how the x’s cancel out

This one is quite difficult

Let

Re-arrange to get

Much easier

Let

The cancel

Indefinite Integration by-parts

Work through these examples
carefully, it is actually a very easy technique once the technical notation has
been resolved.

There are actually only a few examples
which can be asked on an A-level example.The general formula is:The hard part is to know how to set up the
actual formula.

Generally, let be the term which will significantly
simplify the most when differentiated.

Therefore we define:

Remember to do the final integration. Students often forget
this.

Where one needs to be careful is
when one meets the integral of the natural logarithm.

substantially simplifies when differentiated
so:

The trick is to write

substantially simplifies when differentiated
so:

Key integration facts

Even with modern calculator able work
out exact trigonometry expressions in terms of surds, it is nevertheless good
practice to understand and be able to use a number of key trigonometry values.
This is especially the case if you aim to apply to study at university.
University examinations and entrance examinations often rely up this knowledge!

Principle angle facts:

0

(30)

(45)

(60)

(90)

Cosx

1

0

Sinx

0

1

Tanx

0

1

“infinity”

Using symmetry, so you can
use these to work out the exact value of a number of expressions between 0 and
2π.

Definite integration by-parts

We integrated this earlier on.

Remember that the +c is not required for a definite
integrand.

We integrated this earlier.

Also not that

Definite integration by
substitution

See if you can work out an answer for this?

We integrated this earlier on.

The key idea here is to transform the limits using the transformation .